transport through a clay barrier with the contaminant

Transport in Porous Media29: 159–178, 1997. 159© 1997Kluwer Academic Publishers. Printed in the Netherlands.

Transport Through a Clay Barrier with the Contaminant

Concentration Dependent Permeability

MARIUSZ KACZMAREK1,?, TOMASZ HUECKEL1, VIKAS CHAWLA 1,?? andPIERLUIGI IMPERIALI21Duke University, Durham, NC 27708-0287, U.S.A.2ISMES, via Pastrengo, 9, 24068 Bergamo, Italy

(Received: 22 August 1996; in final form: 2 July 1997)

Abstract. Transport of contaminants through clays is characterized by a very low dispersivity,but depends on the sensitivity of its intrinsic permeability to the contaminant’s concentration. Anadditional constitutive relationship for a variable intrinsic permeability is thus adopted leading toa coupled system of equations for diffusive–advective transport in multicomponent liquid. A one-dimensional transport problem is solved using finite difference and Newton–Raphson procedure fornonlinear algebraic equations. The results indicate that although diffusion contributes to an increaseof transport with respect to pure advection, the flux ultimately depends on end boundary conditionsfor concentration which, if low, may actually slow down the evolution of concentration and thus ofpermeability. Indeed, the advective component of flux may still remain secondary if the end portionof the layer remains unaffected by high concentrations. With no constraints on concentration at thebottom (zero concentration gradient boundary condition) and high concentration applied at the top,a significant shortening of the breakthrough time occurs.

Key words: variable permeability, contaminant transport, advection/diffusion, clays, organics, finitedifference.

1. Introduction

Soil liners are major components of containment barriers of waste landfill. Theydetermine an effectiveness of the barriers against the transport of liquid contami-nants into groundwater and, consequently, the safety of the landfill operation. Oneof the important tools used in the design of landfill liners is the numerical simulationof the contaminant transport process. In addition to the prediction of the flux of thecontaminant, it is used to assess the role of different physico-chemical mechanismsinfluencing the transport in the evolution of contamination. Although, the legislativerequirements for liner materials impose limits on hydraulic conductivity only (itsmaximum value is usually limited to 10−6 or 10−7 cm/s), at least four mechanisms:advection, diffusion, dispersion, and sorption control the transport of contaminantsthrough the low permeability barriers (see, e.g., [21, 22, 24]). Moreover, it is exper-imentally well documented that the intrinsic permeability of clayey soils, which are

? Present address: Department of Environmental Mechanics, College of Education, Bydgoszcz,Chodkiewicza 30, 85-064 Bydgoszcz, Poland.

?? Present address: Hibbit, Karlsson and Sorensen, 1080 Main St., Pawtuckett, RI 02860, U.S.A.

160 MARIUSZ KACZMAREK ET AL.

widely used as liner materials, may be substantially altered by some concentratedorganic contaminants, see, e.g., [9, 16]. Microscopic mechanisms responsible for thechanges are of physico-chemical and/or purely chemical nature. Redistribution of thepore space driven by dehydration of clay fraction, and rearrangement of clay parti-cles (flocculation, peptization, and micro-migration) together with chemical reactionsbetween contaminants and clay mineral, such as dissolution of the solids are believedto be the most important causes of the permeability evolution. In the case of linersbeneath domestic landfills, the sensitivity of the liner material to organic contam-inants is especially important. A number of experimental studies have shown thatintrinsic permeability increase in such a case may range few orders of magnitudes.Since concentration may change due to both advection and diffusion, even in thematerials having a very low permeability initially, advection may increase due todiffusive transport resulting in an eventual massive flow. Density and viscosity ofcontaminant–water mixture are also functions of the concentration of contaminant[27]. Finally, concentration and pressure at liner boundaries change due to variabilityof environmental conditions.

The above factors require that the transport processes in liners be approached ascoupled phenomena. Existing models of coupled transport phenomena in saturat-ed porous materials deal with the dependence of mass fluxes on gradients of porepressure, temperature, and concentration of chemicals (e.g. [15]). In the proposedformulation, the dispersive transport depends on advection through mechanical dis-persion, while through the concentration sensitive permeability the advective trans-port depends on diffusion. As a result, a solution of a coupled nonlinear system ofequations is required.

The role of the chemically induced changes of porosity through dissolution orprecipitations of minerals, and as a result of permeability on spatial and temporalevolution of the transport was studied for rock materials, e.g. by Steefel and Lasaga[25], and Carnahan [5]. In these models the central role is played by porosity changes.However, this is not the case of low permeability clayey materials permeated byconcentrated organics, which do not include significant changes in porosity, as amplydemonstrated by Acaret al. [1] and Fernandez and Quigley [8, 9].

The purpose of this paper is to study advective/diffusive/dispersive transport of acontaminant through the chemically sensitive clay layer with intrinsic permeabilitydependent on the concentration, which affects the tiny, bottle neck elements of flowchannels, without much effect on the overall porosity [14]. Thus, the solid skeletonis assumed as undeformable, i.e. with stress and contamination insensitive porosity.The liquid phase is assumed to be a mixture of strongly interacting components [26].This implies that a single balance of linear momentum is considered for the wholeliquid phase, while for its components separate balances for mass conservation onlyare taken into account. Chemical reactions between liquid components and chemicalinteractions of liquid with solid are disregarded. A central point of this approach arethe changes of permeability induced by the chemical dehydration of clay fraction. Theprocess of transport of a contaminant is assumed to be isothermal and slow enough

TRANSPORT THROUGH A CLAY BARRIER 161

to neglect inertial effects. The study is limited to a one-dimensional implementationof the coupled model. Numerical simulation of the process is performed using finite-difference formulation based on backward Euler upwind differencing scheme of thecoupled system of governing equations. The Newton–Raphson algorithm is usedfor solution of the resultant system of non-linear algebraic equations. Although thepresence of concentrated organics in the landfill leachate may be viewed as a limitingcase, it may occur locally in sufficient quantities to cause a critical situation. Also,industrial impoundments may contain significant amounts of concentrated organics.Concentrated dioxane is taken as an example of contaminant in this paper using thedata on its effect on Sarnia clay obtained by Fernandez and Quigley [8–10].

2. Description of the Model

2.1. balance equations

The principle of conservation of mass for the whole mass of liquid implies that therelation between the intrinsic density of liquid,ρ, and the mass average advectivevelocity of the liquid phasev, be expressed as follows:

∂(nρ)

∂t+ ∇ · (nρv) = 0, (1)

wheren is an effective porosity of the liner material. Since most of the soil linermaterials contain clay fraction, a significant amount of water in the materials isadsorbed on the surface of clay particles. As a result, the flow of liquid and advectivecomponent of transport are limited to the pore space which is filled with mobile(nonadsorbed) liquid. The effective porosity,n, represents the volume fraction ofthe mobile liquid. The importance of the effective porosity in modeling of transportphenomena results from the fact that it relates the velocity of advective transport,v,to discharge velocity,q, i.e. q = nv, which is a basic measured quantity for flowthrough porous media.

If the liquid saturating the porous medium is a mixture ofN chemically inertcomponents, then(N − 1) independent mass balances for components, should besatisfied [11], i.e.

∂(nρi)

∂t+ ∇ · (nρivi ) = 0, i = 1, . . . , N − 1, (2)

whereρi is the mass of theith component per unit volume of the liquid (massconcentration of theith component), andvi is the mass average velocity of theithcomponent. The densities and velocities of components and of mixture are relatedas follows:

N∑i=1

ρi = ρ,N∑

i=1

ρivi = ρv. (3)


The transport of liquid through clayey liners is very slow and thus inertial forces andinertial interactions between liquid components are insignificant. Hence, the conceptof strongly interacting constituents of liquid may be adopted [26]. Consequently,the balance of momentum for the considered system reduces to a single equation oflinear momentum for the liquid mixture [11]:

∇ · T + nρb = R, (4)

whereT is stress tensor in the liquid,R the interaction force between liquid and theporous matrix, andb the density of the external volume forces.

2.2. constitutive assumptions

The constitutive hypotheses needed to formulate the problem of contaminant trans-port concern the transport mechanisms and their possible evolution. The basic phe-nomena which control the transport of contaminants through low permeability soilsare: flow of liquid induced by pressure gradient, diffusion of contaminants in theliquid phase driven by concentration gradient, and interaction of liquid and soliddriven by various mechanisms, but usually related to concentration or its gradient,see [21, 22, 24]. In this paper, it is assumed that sorption has negligible effect and canbe disregarded. Advective and diffusive transport are coupled through two mecha-nisms. First, a well established form of coupling occurs through dispersion, whichalthough originally numerically insignificant for clays, depends on fluid velocity.The second type of coupling is typical of clays and is the focal point of this paper. Itis postulated that during contamination, intrinsic permeability is a function of con-centration. A characteristic increase in permeability of clay is observed when porewater is replaced by concentrated organic liquids or water solutions of salts [9, 16].One of hypotheses used to explain it invokes a rearrangement of the pore structureof the clay fraction caused by the collapse of clay clusters due to chemically induceddesorption of water (dehydration). At least two different scenarios of the influenceof clay dehydration on permeability evolution are possible. One of them is basedon the assumption that the dehydration causes significant changes of the volume ofclayey material, leading to a macroscopic fracturing with the fracture size of theorder of milli- or centimeters and a preferential flow of contaminants through thematerial [6]. Another model (following the findings of Fernandez and Quigley [9]) isbased on a hypothesis that the volume changes of clay fraction cause the opening ofmicro-channels with the size of the order of micrometers and less, which control thepermeability while no macro-channels are developed. Both scenarios are observed insoils depending on their fine fraction, mineralogy, water content, contaminant type,and testing techniques. This paper follows the latter idea incorporating a quantita-tive law of evolution of intrinsic permeability [14] to simulate the transport throughclayey soils free of macro-cracks.

The specific assumptions of the model are as follows:(1) Only one-dimensional transport is considered.


(2) The Darcy’s law is formulated indirectly through the seepage forceR betweenthe liquid and the solid entering the linear momentum Equation (4). The force isproportional to the velocity of liquid with respect to solid:

R = µ

Kn2v = γ

κn2v, (5)

whereK is intrinsic permeability of soil,µ is dynamic viscosity of fluid,κ (=Kγ/µ) is hydraulic conductivity, andγ (= ρg) is unit weight of liquid. The valueof coefficient of dynamic viscosity of liquid (mixture of water and contaminant)depends on the type of contaminant and is assumed as a polynomial function ofconcentration related to fluid density,c/ρ [27]:

µ = µwρ

ρw

[µ1

(c

ρ

)3+ µ2

(c

ρ

)2+ µ3

(c

ρ

)+ µ4

], (6)

wherec is concentration (mass of the contaminant per unit volume of the liquid),µw

stands for viscosity of water whileµ1, µ2, µ3 andµ4 are empirical constants.(3) The phenomenological constitutive law for the evolution of intrinsic per-

meability is based on experimental data from the fixed wall, discharge controlledpermeability tests conducted for Sarnia clay, permeated with dioxane with vari-ous concentration by Fernandez and Quigley [9]. These tests have shown an initiallow sensitivity of permeability toward concentration, with a subsequent dramaticincrease in permeability beginning around the mass fraction of contaminant of 0.7(Figure 1). At the same time an evolution of pore space has been observed usingmercury porosimetry, consisting in some changes in size and frequency of occur-rence of dominant pore sizes. It is hypothesized that the permeation of clay with arelatively low dielectric constant organic causes dehydration of clay platelets andconsequent shrinkage of clusters opening larger spaces between them. To simulatethe permeability changes a micro-structural model of pore system conducting liquidhas been proposed [14] as a series connection of larger and smaller channels formedby the dominant pores (Figure 2). The pores correspond to respectively unstressedand stressed contacts between clay domains of randomly oriented clusters of paral-lel platelets [18]. The size of the smaller channels located between parallel stressedinterfaces is found to be critical for the flow discharge, as it results from the assump-tion of series connection of the pores. Intrinsic permeability of an elementary cellcontaining one such channel series is

K = B

1/(δv)3 + 1/4(δh)3 , (7)

whereδh and δv stand for the sizes of smaller and larger channels, andB is aconstant which depends on model parameters (size of elementary cell and tortuosity).Furthermore, it is assumed that chemical dehydration influences only the size of thesmaller channels,δh, the following form of the evolution of this size is proposed:

δh = δh0 + A(eαc/ρ − 1), (8)


Figure 1. Intrinsic permeability of water compacted brown Sarnia clay permeated withleachate-dioxane mixtures. (All tests run originally with reference water,•, followed bypermeants,4, varying from 100% leachate to 100% dioxane) – from [9];σ ′

v0 is the effectivevertical stress ande0 the initial void ratio.

whereA andα are constants to be determined from experimental data. It must beunderlined that the above micro-mechanism is not the only one that can be visualizedin chemically affected clays. Flocculation model, associated with particle rotation dueto edge to face attraction has been seen to simulate well the measured permeabilityand pore system evolution [14]. The actual micro-structural changes seem to be asuperposition of the above mechanisms.

(4) As a consequence of the assumption (3), porosity is unaffected by contami-nation. In fact, the smaller (variable) channels constitute a volume fraction about 50times smaller than the larger ones, which remain constant.

(5) The liquid phase consists of water and a single contaminant miscible in waterwith densityρw andρc, respectively. The volume of the liquid solution is assumed tobe the sum of the volumes of the components taken separately at the same pressure


Figure 2. Pore size change model. (a) Schematic of the structure and flow of water permeatedclay. Single lines correspond to clusters of parallely oriented platelets; (b) Structure and flowof dioxane permeated clay.

(perfect solution, see [20]). Hence, the relation between the density of liquid,ρ, andpartial density of the contaminantc takes a form

ρ = ρw + c

(1 − ρw

ρc

). (9)

(6) The liquid components do not react chemically with the solid or water.(7) The deformations of solid skeleton and the changes of effective porosity of

the soil during the transport are neglected. Since the volume of dehydrated water isextremely small, it is disregarded as a source in mass balance.

(8) The flux of mass of a contaminant,cvc (vc is the mass average pore velocity ofthe contaminant), is decomposed into advective,cv, and dispersive,c(vc − v), parts.The dispersive component of the flux is proportional to the gradient of the partialdensity of contaminant [3]:

c(vc − v) = D̃∇c, (10)

whereD̃ is a coeffecient of dispersion. For one-dimensional case and isotropic porousmedium, the coefficient̃D may be represented as the sum of the effective coefficientof molecular diffusionD and of the term describing mechanical dispersiond|v|:

D̃ = D + d|v|, (11)

whered is a coefficient of longitudinal dispersion. It is assumed that the rearrange-ment of micro-structure, responsible for the evolution of permeability does not influ-ence coefficientsD andd, because they depend mainly on porosity, and characteristic


size of pores of the soil, which in the considered model are not strongly affected bythe contaminant.

(9) The stress tensor in liquid is represented by a product of pore pressure,p, andporosity

T = −np. (12)

2.3. governing equations

The constitutive assumptions introduced in Section 2.2, and the balance Equations(1), (2) and (4), can be reduced to the following system of governing equations:

ρ̄∂c

∂t+ (ρw + cρ̄)

∂v

∂x+ vρ̄

∂c

∂x= 0, (13)

∂c

∂t+ c

∂v

∂x+ v

∂c

∂x− (D + dv)

∂2c

∂x− d

∂c

∂x

∂v

∂x= 0, (14)

(ρw + cρ̄)gK − K∂p

∂x− µnv = 0, (15)

where

ρ̄ = 1 − ρw

ρc. (16)

In this system the only dependent variables arec, v andp. D, d, g andn are constantsandµ andK are known functions ofc.

In general, Equations (13)–(15) constitute a system of coupled nonlinear partialdifferential equations which supplemented with initial and boundary conditions hasto be solved numerically.

2.4. initial and boundary conditions

To investigate the transport of contaminant through clay with a chemically degradablepermeability on engineering scale a one-dimensional diffusive–advective transportof a contaminant is considered through a finite layer. Two sets of simple initialand boundary conditions are analyzed. The initial concentration of the contaminantthroughout the layer is zero. The pore pressure and concentration at the upper bound-ary,pn, andcn, are kept constant. At the lower boundary, the pressure,p0, is assumedto be equal to zero while for concentration at the lower boundary two cases are con-sidered. In the first case it is assumed that concentration is zero, while in the secondcase it is assumed that the gradient of concentration is equal to zero. The condition forconstant concentration at the upper boundary corresponds to a continuous feeding bycontaminant at the boundary. The two conditions at the lower boundary correspond


to a continuous perfect washing and lack of washing of the boundary, respectively(see [28]).

3. Numerical Simulation of Transport Through Liner

3.1. discretization

To discretize the governing partial differential equations, the implicit finite-differencemethod is used. Expansion of Equations (13)–(15) with finite-difference representa-tions of derivatives leads to the following equations

ρ̄

1t(cn+1

j − cnj ) + 1

1x(ρw + ρ̄cn+1

j )(vn+1j+1 − vn+1

j ) +

+ ρ̄

21xvn+1j (cn+1

j+1 − cn+1j−1 − ξcn+1

j−1 + 2ξcn+1j − ξcn+1

j+1 ) = 0, (17)

1

1t(cn+1

j − cnj ) + 1

1xcn+1j (vn+1

j+1 − vn+1j ) +

+ 1

21xvn+1j (cn+1

j+1 − cn+1j−1 − ξcn+1

j−1 + 2ξcn+1j − ξcn+1

j+1 ) −

−(D + dvn+1

j )

2(1x)2 (cn+1j+1 − 2cn+1

j + cn+1j−1 + cn

j+1 − 2cnj + cn

j−1) −

− d

(1x)2 (cn+1j − cn+1

j−1 )(vn+1j+1 − vn+1

j ) = 0, (18)

(ρw + ρ̄cn+1j )gK − K

1x(pn+1

j − pn+1j−1 ) − µnvn+1

j = 0, (19)

where1x is the space grid size and1t is the time step. The superscripts at theindependent variables in Equations (17)–(19) refer to time index and the subscriptsrefer to the space index. The discretization involves using backward Euler method forthe time derivatives, upwind differentiation for the first-order space derivatives andCrank–Nicolson scheme for the second-order space derivatives. Forward or backwarddifferentiation is used for upwind differentiation depending on the direction of advec-tive transport. In Equations (17) and (18), the concentration gradient is approximatedusing second-order upwind differentiation technique, see, e.g., [4, 19], whereby arti-ficial diffusion (terms related to parameterξ ) is added to the advection term. Thisis required to satisfy the accuracy requirements in computation of fluid velocity,vv

[19].The numerical solution of the above problem is obtained by setting up all three

discretized governing equations at each internal point on the space grid. The ini-tial condition is implemented by prescribing initial concentration at all the points


on the space grid. The boundary conditions are prescribed by fixing the values ofconcentration and pressure at the boundaries. Since concentration and pressure atthe boundaries are defined by boundary conditions, while velocities at the bound-aries are unknown, two additional equations are necessary, to determine the bound-ary values for the fluid velocity. The first one imposes the validity of the Darcy’slaw across the upper boundary. At the lower boundary, the extrapolation formulavn+1

2 = vn+11 + 1

2(vn+12 − vn+1

0 ) is used to relate the velocity at the boundary withvelocities at the next two grid points. This gives a system of coupled algebraic equa-tions to be solved for an equal number of unknowns – concentration, velocity, andpressure at each point on the grid. These equations can be expanded and readjustedso that the terms containing explicit unknowns (variables evaluated at the new timestep) are on the left-hand side and the terms explicitly dependent only on the vari-ables evaluated at the previous time step are on the right-hand side. This gives a setof simultaneous algebraic equations which for brevity are written in a matrix formas

Ayn+1 = Byn, (20)

whereA andB are the coefficient matrices depending on the unknowns,yn+1 is thevector of unknowns, andyn is the vector of variables evaluated at the old time step.Due to the nonlinear nature of the considered partial differential equations the systemof algebraic Equations (20) is also nonlinear. The solution to this system is obtained byusing Newton–Raphson algorithm. This methodology involves an iterative evaluationof the incremental solution for each time step until the increments become sufficientlysmall as defined by the tolerance limits. The solution is considered satisfactory whenthe steady state is reached, i.e. when there is no significant change in the solution insuccessive time steps, or when an imposed maximum number of time step iterationsis reached.

3.2. test of the numerical procedure

The above numerical procedure has first been tested against available analyticalsolution for a special case corresponding to constant pore fluid velocity and constantconcentration at the boundaries (see, e.g., [17]) for which constant viscosity andpermeability, and identical mass densities of water and contaminant were assumed.The pressure boundary condition at the upper boundary is 1 m of water, and theconcentration of the contaminant with respect to water density amounts to 0.5. Thepressure and concentration at the lower boundary are zero. The space grid size andthe time step are equal to 0.033 m, and 0.2 yr, respectively. The values of intrinsicpermeability, diffusion and dispersion coefficients are equal to 9.0×10−18 m2, 0.01m2/yr, and 0.0875 mm.

The analytically and numerically obtained distributions of concentration for afew selected time instants are compared in Figure 3. The agreement of the resultsindicates that the applied method of discretization of the governing equations and


Figure 3. Distribution of relative concentration of contaminant (concentration divided bywater density) across the layer. Comparison between an analytical and numerical solutionfor constant viscosity, constant permeability and identical water and contaminant densities.

in particular the effect of artificial diffusion introduced to assure the stability ofnumerical calculations for higher pore velocities do not imply significant deviationof the numerical solutions from the exact ones, even for long times and for transportdominated by diffusion.

3.3. results and discussion

The transport of dioxane through an initially water saturated layer of Sarnia claystudied experimentally by Fernandez and Quigley [8, 9], is considered as an example.The thickness of the layer is taken as 1 m. The values of the material parameters used inthe calculations are as given in Table I. The parameters which define the permeabilityevolution and viscosity of liquid mixture according to formulas (10) and (11) aredetermined using experimental data given by Fernandez and Quigley [8, 9]. Thecoefficients of diffusion and dispersion are typical ones for materials of soil liners.

To illustrate the dependence of constitutive functions on concentration of dioxanethe variation of relative mass density,ρ/ρw, relative viscosity,µ/µwater , of fluid, andrelative intrinsic permeability of soil,K/Kwater , as functions of mass fraction of thecontaminant,c/ρ are shown in Figure 4. The dependence of intrinsic permeability on


Table I. Material parameters for the consid-ered problem.

Material property Value

g 0.976× 1016 m/yr2

ρw 1 × 103 kg/m3

c 1.44× 103 kg/m3

µwater 3.3 × 104 kg/m yrD 0.01 m2/yrd 8.75× 10−5 mδv 3 × 10−6 mδh0 6.7 × 10−8 mB 7.576× 103 m−1

A 2.32× 10−12

α 12.52n 0.3µ1 0.0µ2 −4.08µ3 4.28µ4 1.0

Figure 4. Variation of relative mass density, relative viscosity of fluid and relative intrinsicpermeability of soil as functions of mass fraction of the contaminant in pore fluid followingthe proposed model.×– experimental data for Sarnia clay obtained by Fernandez and Quigley[8, 9].


Table II. Boundary conditions for concentration and pressure.

Case Low concentration High concentration

1 c/ρw = 0.5 t > 0 x = 1 m c/ρw = 1.44 t > 0 x = 1 mc/ρw = 0 t > 0 x = 0 m c/ρw = 0 t > 0 x = 0 mp = 3 ma t > 0 x = 1 m p = 3 m t > 0 x = 1 mp = 0 m t > 0 x = 0 m p = 0 m t > 0 x = 0 m

2 c/ρw = 0.5 t > 0 x = 1 m c/ρw = 1.44 t > 0 x = 1 m∂c/∂x = 0 t > 0 x = 0 m ∂c/∂x = 0 t > 0 x = 0 mp = 3 ma t > 0 x = 1 m p = 3 m t > 0 x = 1 mp = 0 m t > 0 x = 0 m p = 0 m t > 0 x = 0 m

aPressure in m of water head.

Table III. Initial conditions for concentration.

Low concentration High concentration

c/ρw = 0 t = 0 06 x < 1 m c/ρw = 0 t = 0 06 x < 1 mc/ρw = 0.5 t = 0 x = 1 m c/ρw = 1.44 t = 0 x = 1 m

concentration approximating experimental data by Fernandez and Quigley [9] allowsone to introduce rough distinction between regions of weak (c/ρ < 0.7) and strong(c/ρ > 0.7) sensitivity of intrinsic permeability to the contamination. To assessthe role of this sensitivity for transport, the cases of low and high concentration ofthe contaminant at the upper boundary of the liner are considered separately. Thecomplete set of boundary and initial conditions which corresponds to the constantconcentration value at the upper boundary and the two cases of zero concentrationand zero gradient of concentration of contaminant at the lower boundary are listedin Tables II and III, respectively.

Temporal evolution of the profiles of concentration, and liquid pressure throughoutthe layer subjected to low and high concentration at the upper boundary and for thetwo considered conditions at the lower boundary are shown in Figures 5–7. Thedistributions of concentration for the low concentration at the upper boundary andzero concentration at the lower boundary (Figure 5a) reaches a stationary state afterabout 40 yr. For comparison in Figure 5a prediction of steady state given by theanalytical solution is shown. It is clear that the analytical solution (based on linearand uncoupled model) may give only a rough approximation of the solution whichincorporates the variation of mass, viscosity of liquid and the couplings, althoughonly slight changes in permeability occur for the low concentration case.

For the case of the high concentration at the upper boundary and zero concentra-tion at the lower boundary, the evolution of distribution of concentration is given inFigure, 5b. Because of the strong nonlinearity of the equations (a slight variation ofconcentration induces a large change in permeability) the stationary state of concen-


Figure 5. (a) Evolution of distribution of concentration of dioxane across the clay layerfor zero concentration at the bottom boundary and low concentration imposed at the layertop. (b) Evolution of distribution of concentration of dioxane across the clay layer for zeroconcentration at the bottom boundary and high concentration imposed at the layer top.


Figure 6. Evolution of distribution of concentration of dioxane across the clay layer for zerogradient of concentration at the bottom boundary and high concentration imposed at the layertop.

tration is not reached. High concentration of the contaminant spreads over almost thewhole layer already at 40 yr as the result of increasing permeability. It is interestingto note that the velocity of fluid is only fifteen times higher than the initial value ofvelocity, which is much less than the increase of hydraulic conductivity in the majorpart of the layer (hydraulic conductivity increases about thousand times). There aretwo factors which limit the velocity evolution. One of them is the fact that for theone-dimensional problem with the above boundary conditions the layers more andless affected by the change in permeability are arranged in a series system. Sincethe lower part of the clay layer preserves low hydraulic conductivity, by the natureof the series system the advective mode of the transport is strongly attenuated. Thesecond factor, which limits the growth of the advective velocity, is the zero concen-tration boundary condition at the washed lower boundary, which forces a drop ofconcentration and thus prevents an excessive change in permeability at the lowerpart.

The above attenuation of the increase of concentration and velocity is not observedin the case with zero gradient of concentration at the lower boundary. In such casethe penetration of contaminant throughout the layer is significantly accelerated, seeFigure 6, and finally the concentration reaches a homogeneous distribution. Themaximum value of the pore fluid velocity in this case is about thousand times higherthan the initial velocity, which practically means failure of the barrier. In the exam-


Figure 7. Evolution of distribution of pore pressure across the clay layer for zero concentrationat the bottom boundary and high concentration of dioxane imposed at the layer top.

ples corresponding to high concentration of contaminant at the upper boundary, asignificant deviation of pore pressure from the linear distribution is predicted. Inthe part of the layer affected by large changes of permeability, pore pressure raisessubstantially beyond the value prescribed at the upper boundary (see, e.g., results forcase 1 in Figure 7). In fact, according to Equation (5), the interaction force betweenliquid and porous solid, which is the only source of dissipation of energy of fluid, isconsiderably smaller in the region of high concentration than in the rest of material.As a result, to maintain the equilibrium in the region of high concentration, the fluidpressure must increase.

To further investigate the consequences of the chemically driven variations inpermeability at higher concentrations, let us examine the evolution of velocity ofliquid with time (Figure 8). The mass averaged velocity of fluid is a primitive quantityin our model. To measure the volumetric flow across a unit area, we use the volumevelocity of mass fluxes of water and of contaminant divided by the actual densitiesof water and contaminant, respectively. The temporal changes of volume velocityare the result of variation in fluid viscosity and intrinsic permeability of the porousmaterial due to contamination. Initially, the enhancement of advective transport bydiffusion takes place. Then, if the higher concentration at the upper boundary isapplied, a breakthrough mechanism develops. The time of its occurrence seems to bevery sensitive to the value of the imposed concentration at the top boundary and tothe type of the lower boundary condition. For instance, a change in a lower boundarycondition from zero concentration to zero gradient of concentration decreases thetime of the breakthrough occurrence from about 43 to 21 yr (see Figure 8). On theother hand, a small difference in concentration at the influent boundary between 1.4


Figure 8. Evolution of distribution of fluid volume velocity of flow in time across the claylayer, for different concentrations of dioxane imposed at the layer top and different boundaryconditions.

and 1.44 (the values of concentration are referred to the denisity of water) yields ashortening of the breakthrough time from about 54 to 43 yr.

Note also that, although the advective velocity in the considered case of the highconcentration raises significantly (three orders of magnitude), the comparison ofdiffusion coefficient,D, and of parameter describing mechanical dispersion,d|v|,indicates that for both cases (low and high concentration) the mechanical dispersionstill remains the secondary effect in the transport through clay liners. In the latter case,the mechanical dispersion rises to become about 17% of the coefficient of dispersionleading to an additional, but still minor coupling.

The above analysis is performed using a model which takes into account bothadvective and diffusive contributions to the development of contamination. Sincemost regulatory guidelines for construction of liners disregard the diffusive mecha-nism, it is instructive to compare the predictions of the coupled advective–diffusivemodel for zero concentration and zero gradient of concentration at the bottom bound-ary with the predictions of models in which advection is the only transport mech-anism, see Figure 9. The comparison refers to the low concentration case, and twomodels of advective transport are considered. The first one totally disregards thechanges in material parameters (density, viscosity, and permeability), and the otherone neglects only the changes of density [14]. The advective models yield a sig-nificant time delay of the flux of contaminant at the lower boundary and/or a largedeviation of the value of flux of contaminant at steady state, when compared to thecoupled diffusion–advection model.


Figure 9. Mass flux of dioxane at the bottom boundary as a function of time. Low con-centration of dioxane imposed at the layer top. Comparison of the simulations of the cou-pled diffusive–advective transport with: (1) purely advective transport with constant density,viscosity and permeability, and (2) purely advective transport with variable viscosity andpermeability, and constant density of dioxane.

4. Conclusions

The results presented show the influence of two coupled phenomena: diffusion, andadvection on the development of structural damage in clay barriers. In addition tothe usual coupling through dispersion, which is initially insignificant in clays, themodel includes the permeability sensitivity to organics at high concentration, follow-ing earlier findings for clays by Fernandez and Quigley [8] and Acaret al. [1]. Forlow and moderate concentration of contaminant at the top boundary, the diffusivecomponent strongly influences transport. No large changes in permeability occurin the layer. Only an extremely high concentration imposed at the top boundary ofthe layer can cause a significant degradation of material impermeability throughoutnearly the whole layer and a consequent domination of advective mechanism overdiffusive mechanism of transport. It is shown that the propagation of the evolutionof permeability and of the advective velocity is significantly slower if a low con-centration occurs at the layer bottom. From the practical point of view this indicatesthe importance of a good leachate collection system, and an adverse effect of drainclogging. Thus, although in general diffusion contributes to an increase of the massof the transported contaminant through liners, it may control the evolution of con-centration and therefore of advective component of transport through chemicallysensitive materials. The existence of positive feedback between transport of a con-taminant and changes of permeability of a clay should be taken into considerationwhen the material of a liner is selected and a lining system is designed. In particular,


this refers to composite liners in which the upper layer, exposed to high concentrationof contaminants, should possibly be made of a material insensitive to the chemistryof contaminants.

The results presented emphasize the need for the modeling of coupled phenomenain clay materials to determine phenomena which control transport and to ultimatelyallow to predict and possibly control the performance of clay liners. For a betterapproximation of the real life conditions, however, further studies of the coupledproblems including sorption and reaction of contaminants with solid, interaction ofcomponents of liquid, and deformation of the solid material are recommended.

Acknowledgements

The part of the work performed at Duke University has been supported by a grantn. 940294 from ISMES Spa, Italy, within its cooperation agreement with ENEL(Research Center of Brindisi).

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